Let $X$ be a scheme locally of finite type over a sufficiently "nice" base scheme $S$ (nice in sense of reasonable "finiteness conditions", for sake of simplicity let's start as simple as possible and assume $S$ affine with Noetherian ring), and $G$ a $S$-group scheme acting on $X$, ie there exist $S$ morphism $m:G \times X \to X$ satisfying usual compatibility stuff [main reference: D. Mumford's GIT]

Let $f: T \to X$ be any $T$-valued point of $X$. Then we define the orbit of $f$ with respect to this action to be the image of the map $\psi_f: G \times_S T \to X \times_S T$, where the map is defined as composition $\psi_f:=(m \cdot (1_G \times f), pr_2)$.

Question: If one talks in context of GIT about "the orbit", does one assumes implicitly that it carries certain "natural" scheme structure? Which one? And what one considers then literally, the set-theoretical orbit as underlying set (with this hypothetical mysterious scheme structure) or the scheme theoretic closure (see below on clarification about the latter object)?

Note, that one can always associate canonically to an image a scheme theoretic closure, an object with scheme structure induced by smallest quasi-coherent ideal sheaf contained in $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ Y \to f_*\mathcal{O}_ X)$.

In nice enough situations (eg map quasi-compact and quasi-separated) the underlying topological space of this ideal sheaf coincides with the topological closure of set theoretic image.

But well, what is by convention in context of GIT "the orbit" as scheme? Does (maybe in nice situations) the set theoretic image carry a "canonical" scheme structure which one tacitly assumes in literature, or does one by "orbit" mean in this context always the schematic theoretic closure described above, which in turn has intrinsically given scheme structure?

My motivation is based one my observation that in the literature on GIT one uses often the orbit as "existing object", but I nowhere found a profound discussion which scheme structure it should carry, if one seriously want to study it with algebro-geometric methods.

Let $X$ be a scheme locally of finite type over a sufficiently "nice" base scheme $S$ (nice in sense of reasonable "finiteness conditions", for sake of simplicity let's start as simple as possible and assume $S$ affine with Noetherian ring), and $G$ a affine (#edited later, to keep the story simpler) $S$-group scheme acting on $X$, ie there exist $S$ morphism $m:G \times X \to X$ satisfying usual compatibility stuff [main reference: D. Mumford's GIT]

Let $f: T \to X$ be any $T$-valued point of $X$. Then we define the orbit of $f$ with respect to this action to be the image of the map $\psi_f: G \times_S T \to X \times_S T$, where the map is defined as composition $\psi_f:=(m \cdot (1_G \times f), pr_2)$.

Question: If one talks in context of GIT about "the orbit", does one assumes implicitly that it carries certain "natural" scheme structure? Which one? And what one considers then literally, the set-theoretical orbit as underlying set (with this hypothetical mysterious scheme structure) or the scheme theoretic closure (see below on clarification about the latter object)?

Note, that one can always associate canonically to an image a scheme theoretic closure, an object with scheme structure induced by smallest quasi-coherent ideal sheaf contained in $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ Y \to f_*\mathcal{O}_ X)$.

In nice enough situations (eg map quasi-compact and quasi-separated) the underlying topological space of this ideal sheaf coincides with the topological closure of set theoretic image.

But well, what is by convention in context of GIT "the orbit" as scheme? Does (maybe in nice situations) the set theoretic image carry a "canonical" scheme structure which one tacitly assumes in literature, or does one by "orbit" mean in this context always the schematic theoretic closure described above, which in turn has intrinsically given scheme structure?

My motivation is based one my observation that in the literature on GIT one uses often the orbit as "existing object", but I nowhere found a profound discussion which scheme structure it should carry, if one seriously want to study it with algebro-geometric methods.

Let $X$ be a scheme locally of finite type over a sufficiently "nice" base scheme $S$ (nice in sense of reasonable "finiteness conditions", for sake of simplicity let,s start as simple as possiple an assume $S$ affine with Noetherian ring), and $G$ a $S$-group scheme acting on $X$, ie there exist $S$ morphism $m:G \times X \to X$ satisfying usual compatibility stuff [main reference: D. Mumford's GIT]

Let $f: T \to X$ be any $T$-valued point of $X$. Then we define the orbit of $f$ with resp to this action to be the image of the map $\psi_f: G \times_S T \to X \times_S T$, where the map is defined as composition $\psi_f:=(m \cdot (1_G \times f), pr_2)$.

Question: If one talks in context of GIT about "the orbit", does one assumes implicitly that it carries certain "natural" scheme structure? Which one? And what one considers then literally, the set theorical orbit as underlying set (with this hypothetical mysterious scheme structure) or the scheme theoretic closure (see below on clarification about the latter object)?

Note, that one can always associate canonically to an image a scheme theoretic closure, an object with scheme structure induced by smallest quasi coherent ideal sheaf contained in $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ Y \to f_*\mathcal{O}_ X)$.

In nice enough situations (eg map quasi-compact and quasi-separated) the underlying topological space of this ideal sheaf coincides with the topological closure of set theoretic image.

But well, what is by convention in context of GIT "the orbit" as scheme? Does (maybe in nice situations) the set theoretic image carry a "canonical" scheme structure which one tacitly asumes in literature, or does one by "orbit" mean in this context always the schmetic theoretic closure described above, which in turn has intrinsically given scheme structure?

My motivation is based one my observation that in the literature on GIT one uses often the orbit as "existing object", but I nowhere found a profound discussion which scheme structure it should carry, if one seriously want to study it with algebro geometric methods.

Let $X$ be a scheme locally of finite type over a sufficiently "nice" base scheme $S$ (nice in sense of reasonable "finiteness conditions", for sake of simplicity let's start as simple as possible and assume $S$ affine with Noetherian ring), and $G$ a $S$-group scheme acting on $X$, ie there exist $S$ morphism $m:G \times X \to X$ satisfying usual compatibility stuff [main reference: D. Mumford's GIT]

Let $f: T \to X$ be any $T$-valued point of $X$. Then we define the orbit of $f$ with respect to this action to be the image of the map $\psi_f: G \times_S T \to X \times_S T$, where the map is defined as composition $\psi_f:=(m \cdot (1_G \times f), pr_2)$.

Question: If one talks in context of GIT about "the orbit", does one assumes implicitly that it carries certain "natural" scheme structure? Which one? And what one considers then literally, the set-theoretical orbit as underlying set (with this hypothetical mysterious scheme structure) or the scheme theoretic closure (see below on clarification about the latter object)?

Note, that one can always associate canonically to an image a scheme theoretic closure, an object with scheme structure induced by smallest quasi-coherent ideal sheaf contained in $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ Y \to f_*\mathcal{O}_ X)$.

In nice enough situations (eg map quasi-compact and quasi-separated) the underlying topological space of this ideal sheaf coincides with the topological closure of set theoretic image.

But well, what is by convention in context of GIT "the orbit" as scheme? Does (maybe in nice situations) the set theoretic image carry a "canonical" scheme structure which one tacitly assumes in literature, or does one by "orbit" mean in this context always the schematic theoretic closure described above, which in turn has intrinsically given scheme structure?

My motivation is based one my observation that in the literature on GIT one uses often the orbit as "existing object", but I nowhere found a profound discussion which scheme structure it should carry, if one seriously want to study it with algebro-geometric methods.